Let \(\mathbb{R}^{3}\) denote the three-dimensional space. Take two points \(\mathrm{P}=(1,2,3)\) and \(\mathrm{Q}=(4,2,7)\). Let dist (X, Y) denote the distance between two points X and Y in \(\mathbb{R}^{3}\). Let \(\mathrm{S}=\left\{\mathrm{X} \in \mathbb{R}^{3}:(\operatorname{dist}(\mathrm{X}, \mathrm{P}))^{2}-(\operatorname{dist}(\mathrm{X}, \mathrm{Q}))^{2}=50\right\}\) and \(\mathrm{T}=\left\{\mathrm{Y} \in \mathbb{R}^{3}:(\operatorname{dist}(\mathrm{Y}, \mathrm{Q}))^{2}-(\operatorname{dist}(\mathrm{Y}, \mathrm{P}))^{2}=50\right\}\). Then which of the following statements is (are) TRUE ?

(A) There is a triangle whose area is 1 and all of whose vertices are from S.; (B) There are two distinct points L and M in T such that each point on the line segment LM is also in T .; (C) There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.; (D) There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.

Three Dimensional Geometry — JEE Advanced · 2024 · 26 May 2024 · Paper 1 Question with Solution

Let

R^{3}

denote the three-dimensional space. Take two points

P = (1, 2, 3)

and

Q = (4, 2, 7)

.
Let dist (X, Y) denote the distance between two points X and Y in

R^{3}

. Let

S = {X \in R^{3} : (dist (X, P))^{2} - (dist (X, Q))^{2} = 50}

and

T = {Y \in R^{3} : (dist (Y, Q))^{2} - (dist (Y, P))^{2} = 50}

.
Then which of the following statements is (are) TRUE ?

(A)There is a triangle whose area is 1 and all of whose vertices are from S.✓ Correct
(B)There are two distinct points L and M in T such that each point on the line segment LM is also in T .✓ Correct
(C)There are infinitely many rectangles of perimeter 48, two of whose vertices are from S and the other two vertices are from T.✓ Correct
(D)There is a square of perimeter 48, two of whose vertices are from S and the other two vertices are from T.✓ Correct

Answer: (A), (B), (C), (D)

Solution

Let

X = (x, y, z)

. Then:

(X P)^{2} - (X Q)^{2} = [(x - 1)^{2} + (y - 2)^{2} + (z - 3)^{2}] - [(x - 4)^{2} + (y - 2)^{2} + (z - 7)^{2}] = 50

Expanding and simplifying:

6 x + 8 z = 105

S

is the plane

6 x + 8 z = 105

.

Similarly,

T

is the plane

6 x + 8 z = 5

.

(A) Since

S

is a plane, we can find three non-collinear points forming a triangle of area 1. TRUE.

(B) Any two distinct points

L, M \in T

lie on the plane

T

, so the line segment

L M

lies entirely in

T

. TRUE.

(C) The distance between the parallel planes

S

and

T

is:

d = \frac{∣105 - 5∣}{36 + 64} = \frac{100}{10} = 10

For a rectangle with perimeter 48 and two vertices on each plane, if the sides perpendicular to the planes have length 10, then the other two sides have total length

\frac{48 - 20}{2} = 14

. There are infinitely many such rectangles. TRUE.

(D) For a square with perimeter 48, each side has length 12. If two opposite vertices are on

S

and the other two on

T

, the diagonal connecting vertices on different planes has length

122

. The perpendicular distance between planes is 10. Using the Pythagorean theorem in the plane perpendicular to both

S

and

T

, we can verify such a square exists. TRUE.

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · One or more correct options

JEE Advanced · 2024 · 26 May 2024 · Paper 1 · Three Dimensional Geometry

Solution

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